%I A003095 M1544
%S A003095 0,1,2,5,26,677,458330,210066388901,44127887745906175987802,
%T A003095 1947270476915296449559703445493848930452791205,
%U A003095 3791862310265926082868235028027893277370233152247388584761734150717768254410341175325352026
%N A003095 a(n) = a(n-1)^2 + 1.
%C A003095 Number of binary trees of height less than n.
%C A003095 The rightmost digits cycle (0,1,2,5,6,7,0,1,2,5,6,7,...). a(n) is prime 
               for n = 2, 3, 5, ... a(n) is semiprime for n = 4, ... - Jonathan 
               Vos Post (jvospost3(AT)gmail.com), Jul 21 2005
%C A003095 Apart from the initial term a subsequence of A008318. - Reinhard Zumkeller 
               (reinhard.zumkeller(AT)gmail.com), Jan 17 2008
%D A003095 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A003095 S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 443-448.
%D A003095 R. K. Guy, How to factor a number, Proc. 5th Manitoba Conf. Numerical 
               Math., Congress. Num. 16 (1975), 49-89.
%D A003095 R. Penrose, The Emperor's New Mind, Oxford, 1989, p. 122
%D A003095 M. Tainiter, Algebraic approach to stopping variable problems: Representation 
               theory and applications. J. Combinatorial Theory 9 1970 148-161.
%H A003095 A. V. Aho and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/
               doc/doubly.html">Some doubly exponential sequences</a>, Fib. Quart., 
               11 (1973), 429-437.
%H A003095 P. Flajolet and A. M. Odlyzko, <a href="http://algo.inria.fr/flajolet/
               Publications/publist.html">Limit distributions</a> of coefficients 
               of iterates of polynomials with applications to combinatorial enumerations, 
               Math. Proc. Camb. Phil. Soc., 96 (1984), 237-253.
%H A003095 C. Lenormand, <a href="http://www.ai.univ-paris8.fr/~lenormand/I.2_Magmas_Arborescences.pdf">
               Arbres et permutations II</a>, see p. 6
%H A003095 <a href="Sindx_Aa.html#AHSL">Index entries for sequences of form a(n+1)=a(n)^2 
               + ...</a>
%F A003095 a_n=B_{n-1}(1) where B_n(x)=1+xB_{n-1}(x)^2 is the generating function 
               of trees of height <= n.
%F A003095 a(n) is asymptotic to c^(2^n) where c=1.2259024435287485386279474959130085213... 
               - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 27 2002
%F A003095 c = b^(1/4) where b is the constant in Bottomley's formula in A004019. 
               a(n) appears very asymptotic to c^(2^n) - Sum(k=1,infinity, A088674[k]/
               (2*c^(2^n))^(2*k-1)). - Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), 
               Nov 17 2007
%Y A003095 Cf. A038044.
%Y A003095 Cf. A001699, A056207, A004019.
%Y A003095 A143848, A143849. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Sep 03 2008]
%Y A003095 Cf. A137560, which enumerates binary trees of height less than n and 
               exactly j leaf nodes. [From Robert Munafo (mrob27(AT)gmail.com), 
               Nov 03 2009]
%Y A003095 Sequence in context: A128595 A111195 A064006 this_sequence A023362 A138613 
               A090744
%Y A003095 Adjacent sequences: A003092 A003093 A003094 this_sequence A003096 A003097 
               A003098
%K A003095 nonn,easy,nice
%O A003095 0,3
%A A003095 N. J. A. Sloane (njas(AT)research.att.com), R. P. Stanley
%E A003095 Additional comments from Cyril Banderier (Cyril.Banderier(AT)inria.fr), 
               Jun 05 2000